Spacetime geometry, the speed of light, and its surprising slowness

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Einstein begins special relativity with a physical claim that pretty much every physics student feels to be deeply counterintuitive, if not flatly wrong: that every inertial observer measures the same speed of light in vacuum, regardless of the motion of source or observer.

Experimentally it is so.

Keep the ordinary relativity principle, add this invariance of c, and Galilean transformations have to go. Space and time must instead mix through the Lorentz transformations.

Minkowski changes the paradigm in a much deeper way. Rather than treating Lorentz transformations as peculiar rules for clocks and rulers, regard them as the symmetry transformations of a four-dimensional spacetime with interval

ds² = c²dt² − dx² − dy² − dz².

This geometry automatically divides possible separations into timelike, spacelike and null. Null trajectories satisfy

ds² = 0,

so, rearranging, for motion in one spatial dimension, dx/dt = c.

Has abstract geometry somehow manufactured a very specific physical velocity? The number c has already been inserted as the conversion factor between temporal and spatial units. Define x₀ = ct, and the metric becomes

ds² = dx₀² − dx² − dy² − dz².

The null cone then has slope one. Relativists routinely set c = 1. The famous number 299,792,458 metres per second is therefore not a profound dimensionless constant of nature. It reflects our very human and very historical decision to measure distance in metres and time in seconds.

What is profound is that the universe has a Lorentzian causal structure at all. Massive particles have timelike four-momenta, while massless particles satisfy

E² − p²c² = 0.

Hence E = pc, and the propagation speed (dE/dp) of a massless excitation is c. In modern language, massless particles are zero-invariant-mass representations of the Poincaré group and their worldlines lie on the null cone. That tells us why masslessness and null propagation belong together within relativistic physics. It does not tell us, at some deeper level, why our universe possesses Lorentzian spacetime and massless fields in the first place.

Why does light travel at this particular speed?

Note that a dimensional constant can be changed numerically by changing units. What matters physically are dimensionless ratios.

Light is extraordinarily fast on human scales, yet extraordinarily slow on astronomical ones. It takes about eight minutes to cross the Earth–Sun distance, four years to reach the nearest star, roughly 100,000 years to cross the Milky Way, and billions of years to traverse cosmological distances.

Why should atomic, biological, stellar and galactic scales be separated by such enormous ratios?

That is not a question about special relativity. It is a question about the contingent dimensionless constants of our universe: the strength of electromagnetism, the extraordinary weakness of gravity, particle-mass ratios, cosmological parameters and the scales generated from them.

A universe in which light crossed a galaxy in one human second would not have “a larger c” per se. Its dimensionless hierarchy of physical scales would have to be radically different.

There is also a tight connection between mass and the null cone. In relativity, mass is the invariant magnitude of a particle’s energy-momentum four-vector:

E²/c² − p² = m²c²

Massive particles therefore have timelike four-momentum, with positive invariant norm-squared, and follow timelike worldlines; massless particles have null four-momentum and propagate along null directions, at c.

So masslessness does not merely happen to imply travel at the speed of light: within Lorentzian spacetime the two are structurally linked. The deeper, contingent question is why some fields in our universe are exactly massless while others are not.

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'The Moon Through a Quantum Slit: A Tutorial on Decoherence' - ChatGPT

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The Moon Through a Quantum Slit: A Tutorial on Decoherence

“Do you really believe the moon is not there when you are not looking at it?” asked Einstein, not as a joke but as a pointed challenge to the Copenhagen interpretation of quantum mechanics. His question, outrageous on its face, becomes a gateway to deeper understanding when framed in a modern context: what is the quantum state of the Moon, and how does it compare to the far more familiar example of the double-slit experiment with electrons?

1. The Electron: Superposition and Interference

In the classic two-slit experiment, an electron passes through a barrier with two slits and arrives at a screen. If no which-path information is obtained, the electron behaves as if it passed through both slits simultaneously. Its wavefunction can be written as:

ψ(x) = ψ_L(x) + ψ_R(x)

Here, ψ_L(x) and ψ_R(x) represent the amplitudes associated with the electron taking the left or right path, respectively. Because the total wavefunction includes both paths with a definite phase relationship, the probability of arrival at the screen is:

P(x) = |ψ(x)|² = |ψ_L(x) + ψ_R(x)|²

This leads to interference fringes. The key point: the off-diagonal terms in the corresponding density matrix are non-zero, encoding the ability of different parts of the wavefunction to interfere.

2. Decoherence: Tagging the Path (cf. earlier tutorial)

Now suppose we introduce a detector near the slits that reveals which path the electron took. This need not involve a conscious observer — a passing photon that scatters differently depending on the slit will do. The environment becomes entangled with the electron’s path, and we must describe the system using a density matrix.

Before decoherence, the electron is in a coherent superposition, and the density matrix contains both diagonal and off-diagonal terms:

ρ(x, x') = ψ_L(x)ψ_L^*(x') + ψ_R(x)ψ_R^*(x') + ψ_L(x)ψ_R^*(x') + ψ_R(x)ψ_L^*(x')

The cross-terms — the last two in the sum — are responsible for interference. When decoherence occurs due to environmental entanglement, these terms vanish:

ρ(x, x') = ψ_L(x)ψ_L^*(x') + ψ_R(x)ψ_R^*(x')

This is the density matrix of an incoherent mixture. The result on the screen is two overlapping Gaussians — no interference fringes. The electron has gone from a coherent superposition to a statistical ensemble of alternatives.

3. The Moon’s Wavefunction: Before Decoherence

Now consider the Moon. Its quantum state can, in principle, be described by a wavefunction over position:

|Ψ⟩ = ∫ ψ(x) |x⟩ dx

Before any environmental interaction, this state is a pure superposition over all possible locations — an enormous analogue of the electron's pre-interference wavefunction. It contains the possibility (however implausible) of interference between different Moon positions. But this is not merely philosophical: it is exactly what the formalism demands of an isolated system.

If you were to construct a cosmic interferometer (an absurd idea, but conceptually helpful) that could recombine the Moon’s positional components, you might — in this counterfactual universe — see interference patterns between macroscopically distinct locations.

If you could run identically-prepared copies of the Moon through the interferometer!

4. After Decoherence: The Real Moon

But the Moon is not isolated. It interacts constantly with photons, gravitational fields, neutrinos, and the cosmic microwave background. These interactions entangle the Moon’s spatial wavefunction with vast numbers of environmental degrees of freedom. The result is rapid decoherence.

The Moon's reduced density matrix in the position basis becomes:

ρ(x, x') ≈ 0 for |x - x'| > ℓ_D

where ℓ_D is the decoherence length — often far smaller than an atomic radius. This means that the Moon’s wavefunction becomes a statistical mixture of narrow, localised wave-packets — each one a quasi-classical state. The off-diagonal terms responsible for interference have vanished, and with them, any possibility of observing non-classical motion.

This is mathematically and physically different from a coherent quantum superposition. The wavefunction is no longer "wavy" across great distances. It has become a cloud of classical possibilities, each encoded by its own amplitude-Gaussian, each decohered from the others, evolving independently as if in separate worlds or branches.

5. So What’s the Difference?

You might ask: if there’s only one Moon, and we can’t do a million trials like in the electron case, what’s the real difference between pre- and post-decoherence? Isn’t this all semantics?

No — the distinction is real, even if it's experimentally inaccessible. In principle:

• Before decoherence, interference between locations is possible (though fantastically improbable to observe).
• After decoherence, such interference is physically impossible. The phase relations have been irreversibly scrambled into the environment.

The Moon has gone from being “quantum-coherent but unrealistically so” to being “effectively classical,” and this transition has nothing to do with human observation. The universe itself, via its environment, acts as the ever-watchful observer.

6. Conclusion

The Moon and the electron are not as different as they seem. Both obey the same quantum rules. What separates them is not metaphysics, but scale and entanglement. The electron lives in a regime where interference is feasible. The Moon lives in a regime where decoherence is overwhelming.

The density matrix shows us this difference with clarity. Where the electron's matrix has off-diagonal terms — the mark of quantum interference — the Moon's does not. And that is why we see fringes on a screen for the one, and lunar eclipses for the other.

'You really believe the Moon is not there ...?'

 

The Moon and Measurement: Einstein's Question Revisited

“Do you really believe the Moon is not there when you are not looking at it?”

Einstein’s famous quip was no mere rhetorical flourish. It was a technical objection to the implications of quantum mechanics, directed at the Copenhagen view that unmeasured observables possess no definite values. He was objecting not just to philosophical idealism, but to the notion that physical entities as massive and permanent as the Moon could, in any serious sense, lack a determinate position until observed. For Einstein, such an idea was a reductio ad absurdum of quantum orthodoxy.

The technical heart of his concern lies in the quantum treatment of position and momentum. Quantum theory does not assign definite values to these quantities simultaneously. The best one can obtain is a wavefunction or density matrix encoding a probabilistic distribution, constrained by the uncertainty principle. So what, then, is the Moon's quantum state when no one is measuring it?

To sharpen the issue, let us consider a thought experiment: imagine a Moon entirely isolated from its environment — no light, no gravity gradients, no cosmic radiation, no air molecules. A true quantum island. Suppose we measure its position very precisely at time t = 0, localising its wavefunction into a very narrow peak in the position basis. We have collapsed it into something close to a position eigenstate.

From this point forward, if the Moon is truly isolated, it evolves according to the unitary Schrödinger equation. But a position eigenstate is not a stationary state of the free Hamiltonian — it contains a wide spread of momenta. The result is that the wavefunction begins to spread over time. The Moon’s centre-of-mass position becomes increasingly uncertain as its wavefunction expands. This is not unique to the Moon — it is observed in experiments with electrons, atoms, and even large molecules like buckyballs in quantum interference setups. It is the standard behaviour of a delocalised quantum object.

If we now wait long enough (in practice way longer than the age of the universe for an object the Moon's size) — again, ignoring all interactions — and perform a second position measurement, quantum mechanics says we could in principle find the Moon almost anywhere compatible with its initial momentum spread. Perhaps on the far side of the Earth from where it was first observed. This is not classical orbital motion: this is pure quantum uncertainty in the absence of localisation, an indication that like bound electrons, in this scenario the moon does not really orbit classically. In effect, the Moon's wave function jumps on observation (to a new positional eigenstate).

Repeated measurements could reveal positions all around its orbital path, disconnected from any classical trajectory. It is absurd, and yet entirely within the predictive structure of quantum theory — if the Moon is isolated and we would wait long enough.

But of course, it never is. The Moon is bathed in photons from the Sun, bombarded by particles from cosmic rays, and continuously interacting with the Earth’s gravitational field. These environmental interactions entangle the Moon’s quantum state with the rest of the universe. This is decoherence.

Decoherence is the process by which the off-diagonal elements of the Moon’s reduced density matrix — representing quantum superpositions between macroscopically distinct positions — decay rapidly. The key result from decoherence theory is that such superpositions do not persist for large systems. The Moon’s enormous mass and surface area make it highly susceptible to environmental measurement. Even photons from the cosmic microwave background — with energy on the order of microelectronvolts — suffice to localise its position in femtoseconds.

If you model the Moon as a sphere of radius 1,700 km exposed to the 2.73 K CMB, you can estimate that over 10³⁰ photons strike it every second. Even if only a minuscule fraction scatter coherently, the decoherence timescale for a 1 cm position superposition is vanishingly small: 10^–20 seconds or less. And that is the most conservative estimate, not including solar photons, infrared thermal emission, and gravitational interaction with the Earth. The Moon is, in quantum terms, being continuously measured by the universe.

This constant decoherence dynamically selects a preferred basis — the so-called pointer states — which are robust under environmental monitoring. These states are highly localised in both position and momentum: quasi-classical states. The result is that the Moon appears, and indeed behaves, as though it always has a definite position and trajectory. Decoherence does not require human observers, nor does it invoke collapse. It merely shows that the rest of the universe acts as a measuring apparatus.

Einstein’s rhetorical question still stands, but it has a modern answer. Yes, the Moon is “there” when we are not looking — not because quantum mechanics gives it a determinate position by fiat, but because the environment ensures its continual localisation. The Moon does not jump, because the cosmos is watching.

"I have already been absent, non-existent"

Jenni Diski - writer

I thought this Jenni Diski (1947-2016) article worth noting. Here's an excerpt.

" I am appalled at the thought, suddenly, that someone at some point is going to tell me I am on a journey.

"But much as I hate it, the journey – that deeply unsatisfactory, often deceitful metaphor – keeps popping into my head. Like my thoughts about infinity, my thoughts about my cancer are always champing at the bit, dragging me towards a starting line.

From ignorance of my condition to diagnosis; the initiation into chemotherapy and then the radiotherapy; from the slap of being told that it’s incurable to a sort of acceptance of the upcoming end. From not knowing, to "knowing", to "really" knowing; from being alive and making the human assumption that I will be around "in the future", to coming to terms with a more imminent death. ...

"The end of the 'journey' doesn’t come until you either die cancer-free of something else, or die of the effects of a regeneration of the cancer cells. Good and bad; from here to eternity, and from eternity to here.

"But I have been not here before, remember that. By which I mean that I have been here; I have already been at the destination towards which I’m now heading. I have already been absent, non-existent.

"Beckett and Nabokov know:

I too shall cease and be as when I was not yet, only all over instead of in store.

From an Abandoned Work

The cradle rocks above an abyss, and common sense tells us that our existence is but a brief crack of light between two eternities of darkness.

Speak, Memory

"This thought, this fact, is a genuine comfort, the only one that works, to calm me down when the panic comes. It brings me real solace in the terror of the infinite desert. It doesn’t resolve the question (though, as an atheist I don’t really have one), but it offers me familiarity with:

“The undiscovered country from whose bourn/ No traveller returns.”

"I’ve been there. I’ve done that. And it soothes. When I find myself trembling at the prospect of extinction, I can steady myself by thinking of the abyss that I have already experienced. Sometimes I can almost take a kindly, unhurried interest in my own extinction. The not-being that I have already been."

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Jenni Diski's insight here is real, but for those who know some physics a deeper consolation (perhaps) is that our lives persist in spacetime, a consequence of Einstein's great discovery which I wrote about in my sciencefiction.com article "Sub Specie Aeternitatis".

Confessions of an eternalist

Confession: I am an eternalist.

Eternalist Sean Carroll has a post up commenting on recent books by Lee Smolin and philosopher Roberto Mangabeira Unger who argue to the contrary. Reading the comments there is depressing  - people either don't understand the issue at all, or don't appreciate how to think about the subjective nature of 'now'. Physicists!

One physicist, Sabine Hossenfelder, has written an excellent and definitive analysis: here's an extract.

"The decisive ability that allows us to experience the present moment as being unlike other moments is that we have a memory. We have a memory of events in the past, an imperfect one, and we do not have memory of events in the future. Memory is not in and by itself tied to consciousness, it is tied to the increase of entropy, or the arrow of time if you wish. Many materials show memory; every system with a path dependence like eg hysteresis does. If you get a perm the molecule chains in your hair remember the bonds, not your brain.

"Memory has nothing to do with consciousness in particular which is good because it makes it much easier to find the flaw in the argument leading to the problem of now.

"If we want to describe systems with memory we need at the very least two time parameters: t to parameterize the location of the particle and τ to parameterize the strength of memory of other times depending on its present location. This means there is a function f(t,τ) that encodes how strong is the memory of time τ at moment t. You need, in other words, at the very least a two-point function, a plain particle trajectory will not do.

"That we experience a “now” means that the strength of memory peaks when both time parameters are identical, ie t-τ = 0. That we do not have any memory of the future means that the function vanishes when τ > t. For the past it must decay somehow, but the details don’t matter. This construction is already sufficient to explain why we have the subjective experience of the present moment being special. And it wasn't that difficult, was it?"

I'm not so much into perms as my preferred physical model of subjective temporal experience. If you imagine a robot with an updating model of observed-reality obtained from its internal and external sensors, it's easy to see that examining its own memory store at any instant at all the robot can persuade itself that this instant "now" is special. But all introspective moments are like that. If we saved each database-state where:

"the robot notes that it is aware of itself and its environment now"

to disk, there would be nothing fundamentally distinct about any of them.

That robot is, functionally, me and you and sits easily in the block universe.

Einstein's view of death

Albert Einstein wrote the following, in a letter of condolence to the sister and son of his long-time closest friend, Michele Besso, upon his death, four weeks before Einstein's own (18th April, 1955).

"Now he has departed from this strange world a little ahead of me. That means nothing. People like us, who believe in physics, know that the distinction between past, present and future is only a stubbornly persistent illusion."

Thinking of oneself four-dimensionally is challenging. We think the phrases 'past self' and 'future-self' are metaphors, but actually they are literally true. Somewhere 'up-time' is the future you and me, just as real as myself at this moment of writing, or your moments of instantaneous reality as you read these words.

My past self-slice can communicate with a future self-slice through use of media. If I read this blog later, that's exactly what will happen. I try to imagine that future self, but it's hard. It's not a dialogue: my future self can't answer me back.

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Note added on December 10th 2025

Hi, it's your future self just re-reading this post. I feel quite real I have to say but it's been a while - nineteen and a half years. You'd be pleased to know it turned out pretty well.

Up-time from here is the now-more-imminent moment of my own death.

I sure hope that guy is having an easy time of it - but he can't tell me...

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